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The line x cos alpha + y sin alpha +y...

The line x cos `alpha + y sin alpha +y sin alpha =p` is tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1.`if

Text Solution

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The correct Answer is:
`a^(2) cos^(2) alpha+ b^(2) sin^(2)+ alpha= p^(2), ((a^(2) cos alpha)/(p),(b^(2) sin alpha)/(p))`
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Knowledge Check

  • The line x cos alpha + y sin alpha = p is a tangent to the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 if

    A
    `a^2 cos^2 alpha + b^2 sin^2 alpha = p^2`
    B
    `a^2 sin^2 alpha + b^2 cos^2 alpha = p^2`
    C
    `a^2 cos^2 alpha - b^2 sin^2 alpha = p^2`
    D
    `a^2 sin^2 alpha - b^2 cos^2 alpha = p^2`

  • If the line x cos alpha + y sin alpha = p , is tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) = 1, then the value of a^(2) cos^(2) alpha+b^(2) sin^(2) alpha is

    A
    p
    B
    `p^(2)`
    C
    `(1)/(p^(2))`
    D
    none of these

  • If the line y cos alpha = x sin alpha +a cos alpha be a tangent to the circle x^(2)+y^(2)=a^(2) , then

    A
    `sin^(2)alpha=1`
    B
    `cos^(2) alpha=1`
    C
    `sin ^(2)alpha = a^(2)`
    D
    `cos^(2) alpha=a^(2)`

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    The portion of the line x cos alpha+y sin alpha=p intercepted by the ellipse- (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 subtends a right angle at the centre of the ellipse.Prove that the line touches a circle concentric with the ellipse.

    If the line x cos alpha+y sin alpha=p touches the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 ,then the point of contact will be ((-a^(2)cos alpha)/(p),(-b^(2)sin alpha)/(p)) ((b^(2)cos alpha)/(p),(a^(2)sin alpha)/(p)) ((b^(2)sin alpha)/(p),(a^(2)cos alpha)/(p)) ( (a^(2)cos alpha)/(p),(b^(2)sin alpha)/(p))

    If the line x cos alpha+y sin alpha=p touches the ellipse x^(2)/a^(2)+(y^(2))/(b^(2))=1 then the point of contact will be

    If the portion of the line xcos alpha + y sin alpha = p intercepted by the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 subtends a right angle at the centre of the ellipse, then the line touches a cirlce of radius ab//sqrt((a^2 + b^2)) concentric with the ellipse.

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